C&Pa Learning Focused Culture: Pāngarau/Mathematics 2

March 12, 2022

C&Pa Learning Focused Culture: Pāngarau/Mathematics 2

Our mathematics content focus is Place Value.

When you get to planning your second session, consider your quick image choice carefully and decide what your mathematical big idea focus will be based around your knowledge of the mathematics within the image itself and your knowledge of the ākonga strengths and capabilities. You may decide to keep your big mathematical idea focus the same for your second session using a different image

Grouping for Learning

Grouping for learning, Status and Equity, Facilitating Group Work

Learning Outcomes:

· Identify and describe strategies for organising group work

· Link features of effective group work to the orchestration of mathematical inquiry

· Critically evaluating resources and learning experiences that can be used in group work activity

Effective learning communities are those that provide ākonga with opportunities to work alone and/or with others. In working with others, the kaiako can group ākonga in small groups or in a whole- class situation, or use a mixture of both.

Why do we group ākonga?

Before we look at the ‘how’ of grouping you need to be clear about the purpose of grouping ākonga. Reflect for a moment about how you learn best—in a group, with a peer, or by yourself? At different stages of your life, and for different topics or tasks, you might give different answers.

Grouped together, ākonga of similar achievement levels can be given tasks that target their specific learning pathways – this instructional practice is commonly referred to as ‘ability grouping’ or ‘homogeneous grouping’.

A second reason for grouping is to enable ākonga to work together, to engage in collaborative mathematical inquiry practices.

Anthony and Walshaw (2007). Effective pedagogy in mathematics/pāngarau best evidence synthesis (BES). Wellington: Ministry of Education. The BES section on Working in groups (pp. 64 – 68) looking for the claims that research makes about how groups can support learning.

How should groups be organised?

Effective teachers often use different grouping arrangements depending on the desired learning outcomes associated with tasks.

Higgins & Eden (2015) Practice-based inservice teacher education: Generating local theory about the pedagogy of group work. In Mathematics Teacher Education and Development, 17(2).

As the article by Higgins and Eden suggests, in New Zealand primary classrooms mathematics teaching is often with small groups of ākonga however this is not necessarily the most productive approach

What if…..?

Randomised Grouping:

Liljedahl, P. (2014). The affordances of using visibly random groups in a mathematics classroom. In Transforming Mathematics Instruction (pp. 127-144). Springer, Cham.

What if the setting of groups was left to chance? What if, instead of planned grouping, the assignment of groups was done randomly? Here are some of Liljedahl’s findings when implementing randomised grouping:

students became agreeable to work in any group they were placed in,
the elimination of social barriers within the classroom,
an increase in the mobility of knowledge between students,
a decrease in reliance on the teacher for answers,
an increase in both enthusiasm for mathematics class and engagement in mathematics tasks.

Learning to see strengths and capabilities:

Skinner, A., Louie, N., & Baldinger, E. M. (2019). Learning to See Students’ Mathematical Strengths. Teaching Children Mathematics, 25(6), 338-344.

Firstly reflect on what does being “good at math” look like and sound like to you? How could you expand your definition to include more of the diverse strengths that support mathematics learning—and to include more students?

Now consider the list of strengths that the teacher in this article brainstormed with her students,

Explain how you are thinking.
Listen to new ideas.
Visualize in lots of ways.
Represent (show) your thinking.
Make connections.
Try ideas.

If this is what you value as a teacher, what are the implications for you and what impact might this have on the mathematical learning, engagement and participation of ALL children in your class?

Grouping: Status and Equity

Boaler’s (2008) study where ākonga work in mixed ability groups on maths tasks introduces the concept of ‘relational equity’. According to Boaler, focusing on relational equity in the classroom creates another dimension to the focus on achievement, “that draws attention to the ways students learn to treat each other and the respect they learn for people from different circumstance to their own” (p. 165). Boaler claims that within the mathematics lesson, group work can promote:

respect for other people’s ideas;
commitment to the learning of others; and
learning methods of communication and support.

Shah and Crespo (2018); in this article the authors describe how ākonga who fall under deficit narrative perspectives are often excluded from participating in mathematical discussions and learning and are therefore further marginalised.

These are categorized as Control, control over participation and control over whether the students are correct when they participate, Competitiveness, “competitive and individualistic classroom interactions” (p.30) and Language Proficiency, marginalization related to English language proficiency.

Collaborative group work demands a range of social and cognitive learning behaviours; students must pose numerous questions and conjectures, engage in conflict resolution, and revise their thinking. Students need to understand that they must take responsibility for contributing, active listening, and sense-making.

Hunter, R. (2007). Scaffolding small group interactions.

Key to productive group work:

In summary we will review three important aspects of collaborative group practice

1. Creating a safe, supportive learning environment

How then do you as a teacher negotiate the norms and obligations regarding participation, risk taking, and positioning of student contributions within group work? For this to happen, students need a safe, supportive learning environment that promotes social and intellectual risk-taking. In attending to students’ ‘mathematical relationships’ teachers and students should engage in explicit discussion about their participation rights and obligations concerning contributing, listening, and valuing; that is, there needs to be a common understanding of what it means to be accountable to the learning community.

First and foremost is the requirement to establish conditions for respectful discourse. Discourse is respectful when each person’s ideas—be it student or teacher’s—are taken seriously. Respectful discourse is also inclusive; that is, all contributions are valued and no one person is disregarded. To create such an environment, teachers need to ensure that everyone is willing to contribute and that others will listen carefully. In Hunter’s (2007) study of primary New Zealand classes she found that expectations were most effective when established through negotiation with students. Examples of negotiated rights and responsibilities included:

the right of all students to contribute and to be listened to;
the right to test out ideas that may or may not be correct without fear of having other students making disrespectful comments; and
the right to have other people discuss your ideas and not you.

Socio Group Participatory Norms are an important aspect of creating a safe, supportive, inclusive and productive learning environment

Respectful Communication: Expressing agreement and disagreement

Leaving no one behind: Making sure all members of the group are making sense of the mathematics.

Take no passengers: Ensuring group members are participating and contributing

Developing shared understandings: Ensuring all members of the group are understanding the mathematical thinking.

Asking questions: To seek clarification so they can understand and make sense of each others thinking.

Active listening and contributing: Proper engagement, listening to each other and contributing their ideas and thinking to develop shared understandings.

Collaboration: Working together to productively struggle and persevere to solve the problem or task.

Valuing contributions: Valuing each others contribution to the group work, valuing ALL students contributions.

Equal status: Everyone has something worthwhile to contribute to the group, no one has a high or a low status in terms of being a mathematician.

2. Supporting students to take risks

In rich tasks, where many different ideas draw on a range of mathematical concepts, in-depth discussion provides students with opportunities to extend their understanding of ideas or solutions methods. However, having to share ideas publically involves taking risks. Open-ended activities and modelling type activities, in particular, involve putting ideas out there, tossing ideas around, making conjectures, and sometimes going down the wrong track.

3. Supporting students to be positioned competently

In addition to reaffirming the participation rights and obligations of group activity, the teacher can take a proactive role in highlighting contributions from less able or less vocal students, making sure that their thinking is positioned as valuable.

As a teacher you also need to take care to ensure that academically productive talk is not just for those students with strong verbal skills or for those students who are confident about speaking out. In some groups there are individuals or groups of students who might initially prefer to remain passive. For example, Pasifika girls may be reluctant to speak up or to question a boy’s thinking. Without affirmative support they are likely to be more comfortable in the role of listening respectfully to the teacher. In Hunter’s (2007) research, a proactive comment provided by the teacher towards a Pasifika girl’s contribution was: “You don’t have to whisper. You can talk because we want to make sure that you are heard.” On another occasion a student was told to “speak up, I like the way you are thinking but we need to hear you”.

Anthony, Hunter, & Hunter (2018). How should we group students in primary maths classrooms? [Blog post]. Retrieved from https://nzareblog.wordpress.com/ 2018/02/19/grouping-primary-maths/

Boaler, J. (2008).Promoting ‘relational equity’ and high mathematics achievement through an innovative mixed ability approach. British Educational Research Journal, 34(2), 167-194.

Hunter, R. (2007). Scaffolding small group interactions. In J. Watson & K. Beswick (Eds.), Mathematics: Essential research, essential practice (Proceedings of the 30th annual conference of the Mathematics Education Research Group, Vol. 2, pp. 430-439). Adelaide: MERGA.

Shah, Niral, and Sandra Crespo. “Cultural narratives and status hierarchies: Tools for identifying and disrupting inequity in mathematics classroom interaction.” Mathematical discourse that breaks barriers and creates space for marginalized learners. Brill Sense, 2018. 23-37.

Content: Place Value

Whole Number: Place Value

Learning Outcomes:

Describe the place value system
Count in te reo Māori
Describe suitable activities and equipment for helping tamariki develop an understanding of place value.

Read: Van de Walle, Karp, & Bay-Williams (2015) Chapter 10: Developing whole- number place-value concepts (pp. 246- 274) This chapter presents a comprehensive description of the way understandings of place value acts as an anchors for number relationships. As ākonga develop computational strategies (and part-whole number knowledge) they are enhancing their understanding of place value.

Read: Walls, F., (2010). Handling numbers. (pp. 27-40) in Averill & Harvey (2010) In this article the writer suggests that kaiako should develop children’s understanding of large numbers as metaphors. As you read through this chapter stop and think about the sections termed ‘Points to Ponder’.

The Place Value System:

Understanding our numeration system means understanding place value. Our numeration system is like a code in which the value of a digit is determined not only by its face value but also by the position that it is in. So, although 72 and 27 have the same digits, they do not represent the same number because the digits are in different places. For tamariki to understand the numeration system they have to understand four fundamental properties. These are:

1. A digit has place value, that is, not only the face value of the digit but also its position is important. The total value that the digit represents in a given numeral is the product of its face value and its place value.

2. The base of our number system is 10. That means that each place in the numeration system is 10 times greater than the place to the right of it. It also means that we can write a numeral for any number using just the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

3. The symbol for zero represents the absence of something. It means that other digits retain their correct place value.

Our system is an additive system. That means the total value of the numeral 234 is 200 + 30 + 4. The first is written in compact form as 234 and in expanded form as 200 + 30 + 4.

Place value models fall into the following categories. As shown in the diagram, the categories have been placed in the order in which children should meet them:

Discrete objects that can be grouped into groups of 10.

Discrete objects that can be fastened together.

‘Ready made’ apparatus with 10 objects already joined together but still visible.

‘Ready made’ apparatus representing 10 objects in size but without the objects marked separately.

Apparatus where the 10s and 1s (or 100s) are not proportional in size but are distinguished only by position or shape or colour.

Explore the following equipment animations from the NZMaths website:

Arrow cards
Beans
Place value houses

Using the models:

Discrete material such as beans or counters are suitable equipment for tamariki to use when first modelling the number system. Tamariki should begin with simple activities that require them to count out separate items.

At the early stages the tamariki do not see 10 as a unit of any kind. The aim is for the tamariki to experience counting and making collections so that they begin to see 10 as a new unit composed of ten ones. Activities should be carefully selected to develop and then reinforce the concept of place value.

In the early stages, the experience of putting objects into groups of ten should be provided for ākonga by using a variety of equipment such as Tens Frame and counters, beans, bundling sticks, Towers of Ten, and Place Value or Dienes blocks (poro-uara tū).

The use of Arrow cards (kāri pere), Hundreds Boards/Charts (paparau), Thousands Books, and Number Expanders support later stages in the development of place value understandings.

Understanding the place value system:

It is only through a good understanding of the place value system that tamariki can convert numbers from words to symbols and vice-versa. Without this understanding the number ‘seventy-three’ could quite reasonably be written as 703, or three hundred and twenty seven could be written as 300207 or maybe 30027.

Compact and Expanded Numerals: Compact form is the way we usually write numerals, e.g. 4 298. However, 4 298 can also be written as 4000 + 200 + 90 + 8. This is an expanded numeral. The ability to write numerals in both compact and expanded forms shows a sound understanding of place value.

Arrow cards are another resource that can show how the place value system works.

Using te reo Māori to develop place value understanding

The English words for the numbers whose numerals are 11, 12, 13, 14, 15, 16, 17, 18 and 19 do not strongly indicate the ten plus one, ten plus two etc nature of the numeration system. From twenty onwards it becomes a lot easier. Hence virtually all the English forms of the numerals from one to twenty have to be rote learnt. The Māori number names have a much more obvious base ten foundation and are more helpful through the teens. This is also reflected in counting systems in other cultures.

One Kotahi or Tahi
Two Rua
Three Toru
Four Whā
Five Rima
Six Ono
Seven Whitu
Eight Waru
Nine Iwa
Ten Tekau
Eleven Tekau mā tahi
Twelve Tekau mā rua
Thirteen Tekau mā toru
Fourteen tekau mā whā
Fifteen tekau mā rima
Sixteen tekau mā ono
Seventeen tekau mā whitu
Eighteen tekau mā waru
Nineteen tekau mā ika
Twenty Rua tekau
Twenty rea tekau ma tahi

E hia nga mea? How many

Extending the number system:

The base ten place value system means that the position of a number represents its value.

There is a variety of equipment and activities that can be used to support the understanding of large numbers.These include Thousands Book (Material Master 4–7), Place Value Houses (Material Master 4–11), and Arrow cards (Materials Master 4–14).

Numeracy activities that may help tamariki develop an understanding of large numbers can be found in Book 4: Teaching Number Knowledge of the Numeracy Development Project books.

Task: There are many poems and storybooks that support extended number knowledge for tamariki , can you recall any from your own childhood, do a literature search for stories that you could use to reinforce and extend number knowledge with tamariki.

3.3. Mathematical Tasks and Questioning

Group Worthy Tasks:

Learning Outcomes:

Explain the role of context and other characteristics of rich (worthwhile), group-worthy mathematics tasks.
Describe ways to use learners cultural funds of knowledge and lived experiences as contexts for designing and using rich (worthwhile), group-worthy mathematics tasks.
Explain the role of teachers’ questions in supporting students’ learning in mathematics.

We are talking about worthwhile, group-worthy tasks that can be used interactively with a diverse group of learners. These tasks are high ceiling, low floor tasks with multiple entry points for a range of learners to enter at. (Boaler, 2016)

Read: Van de Walle, Karp, & Bay-Williams (2015) pp. 61-72: Tasks that promote problem solvingin chapter 3.

Context of tasks:

Read the article ‘Using culturally embedded Problem-Solving tasks to promote equity within mathematical inquiry communities’ by Hunter and Hunter (2018), the authors highlight the importance of context and suggest that connecting problems with the rich, lived experiences of Pāsifika ākonga repositions their culturally located funds of knowledge as valuable

The mathematics of home is valued and when classroom tasks are designed with these familiar contexts, ākonga recognise that mathematics exists in their social and cultural lived experiences outside of school. Maths is no longer just school maths but it is a valued part of the everyday lives of our ākonga.

Watch this video and consider the importance of really knowing your learners and their worlds in order to promote authentic sense-making in mathematics.

Rich Task Design:

There are five common strategies for designing a rich task:

Turning the question around – e.g., changing:

“What is half of 20?”

“10 is a fraction of a number, what could the fraction and number be?”

Asking for similarities and differences – e.g.,

“How are the numbers 85 and 100 alike or different?”

Asking for a number sentence – e.g. to describe a dot pattern

Changing the question type e.g., from a join problem to a comparison problem

Replacing a number with a blank – e.g., changing:

“We issued 5 books from the library on Monday then on Tuesday we issued 3 more, how many books do we have from the library?”

“We issued some books from the library on Monday then on Tuesday we issued 3 more, how many books might we have from the library?”

Can you think of other possibilities?

Consider the following three tasks:

1) Solve 6 x 4 =

2) Which family does the fact 6 x 4 = 24 belong?

3) Describe the pattern below by using mathematical equations.

x x x x

What do the tasks have in common?

How is task 3 more rich than the others – what characteristics of a rich task does it have?

What sorts of classroom conversations would be possible from task 3?

Complete each of the following tasks and think about why these are examples of rich tasks. What choices do the ākonga have when they approach the task?

You divide two numbers and the answer is 2.5. What two numbers might you have divided? What word problem might you have been solving?

Fill in values for the blanks to make this statement true: 72 is % of

You add two fractions and the sum is 9/10. What could the fractions be?

Mathematics Tasks and Questioning:

When we consider how teachers can effectively use mathematics tasks, we make links with our previous work on the importance of communication in the mathematics classroom.

Read in the module two folder: Hallman-Thrasher & Spangler (2020). Purposeful Questioning with High Cognitive–Demand Tasks

This article describes how kaiako can develop and pose questions with specific purposes; engage ākonga in tasks to target thinking, communicate and justify ideas and maintain cognitive demand. The authors describe four different question types:

elicit thinking
generate ideas
clarify/revise explanations
justify claims

In the table on page 454, they provide specific strategies and examples for the different question types.

Watch in the module three folder the video, take note of the questioning strategies and the specific language used by the ākonga. The article poses questions for reflection as you watch the video.

Task: In this blog we learn about what are rich tasks in math and why they are important. As you read this blog, critically analyse what the author is saying about differentiation through rich tasks, what do you agree with and what do you disagree with? Be sure to justify why by connecting to literature and think in terms of the math’s material we have explored thus far. You might like to record your critique in a blog post in your portfolio.

3.4. Planning for an effective task launch

An effective task launch:

Learning Outcomes:

Describe the importance of the problem launch for productive student engagement;
Design and implement an effective launch for a complex problem
Design a problem solving lesson with consideration of the key elements of a lesson.

Introduction: Read: Van de Walle, Karp, & Bay-Williams (2020) Chapter 4, pages 81-94 of the recommended text paying particular attention to the before stage of the lesson but also considering how this stage fits with the during and after stages.

Enacting ambitious mathematics teaching: The role of the launch in the context of a lesson.

Read: Hunter et al. (2018). Developing mathematical inquiry communities: Enacting culturally responsive, culturally sustaining, ambitious mathematics teaching.

developing mathematical inquiry communities (DMIC) approach

The approach is premised on the idea that “students need multiple opportunities to engage in rich mathematical talk and challenging tasks” (p. 25). DMIC lessons typically involve:

Design a rich, group-worthy problem task built around a big mathematical idea – including kaiako solving the problem themselves and anticipating students’ responses establishing social group participatory norms related to communication and participation launching the problem ākonga solving the problem in small collaborative groups – with one pen and one piece of paper large group sharing whereby selected groups present their explanations and justifications connecting and generalising – eliciting and analysing connections between strategies and to the big mathematical idea

Why the effective launch of a problem is crucial?

Problem-Launch-Cue-Sheet Download

Consider this example of a problem, how might you set up your ākonga to engage productively with it?

Junior’s Haircutting

At Junior’s haircutting ceremony, there were 64 donuts on the head table. The guests ate a quarter of the donuts and took the rest home to share with their families. How many donuts were taken home to share after the ceremony?

Launch Question 1. What is the story in the problem? What is the context of the problem?

Challenging tasks often involve real-world settings selected to support the development of ākonga reasoning and communication about particular mathematical thinking. Some ākonga may find it challenging to start because they are not familiar with the context, or setting.

Therefore it is important that discussion takes place about any potentially unfamiliar features of the task. For example, the ākonga might be asked to imagine that they are participants in the context and to share what they know about it.

Kaiako may also connect the setting or context to a person, place, or event that might be familiar or of interest to ākonga. What is important for an effective launch is that kaiako do not just talk to ākonga about the story and context of tasks but instead solicit input from the ākonga. Where a context is unfamiliar to some ākonga, this can be an opportunity to call on the expertise of those ākonga for whom the context is well known. For instance, a problem about food preparation for a hair cutting ceremony in the Cook Islands can draw on the expert knowledge of a group of ākonga and represent an opportunity to have their experiences valued in the classroom. The role of the launch is to draw on this expertise and share the knowledge other ākonga would need to understand and engage with the problem.

What is the story in the problem “Junior’s Haircutting”? What different experiences may children have to make story and context familiar or unfamiliar? Who might the “experts” and “learners” be in relation to understanding this story or context? How might you elicit and develop ākonga understandings of these features?

Launch Question 2. What is the problem asking us to do?

Focusing only on the story and the context is not sufficient, it is also important that teachers discuss what the problem is asking without hinting at particular methods or procedures that should be used to solve the task.

When considering what the problem is asking students to do, it is essential that teachers complete the tasks themselves in the planning stage, and anticipate a range of ākonga mathematical understandings and misconceptions. It is important to identify the key understandings ākonga will need to be able to engage in the task, including those that may be unfamiliar. Kaiako need to consider how they will elicit and develop ākonga understandings of these ideas.

What is the problem asking the students to do? What are the key mathematical ideas in the problem “Junior’s Haircutting”? Which if these ideas might be unfamiliar to ākonga? How might you elicit and develop ākonga understandings of what the problem is asking without leading them to a solution?

Develop common language to describe the story and what the problem is asking.

It is important that kaiako do not simply talk to ākonga about the key features of tasks but instead expect ākonga input by asking questions that require more than yes or no responses. This helps the kaiako to determine the level of support the ākonga need to engage successfully in the task. They need to build on ākonga contributions and both support and press them to develop a common language to describe the key features of the task that might be new or confusing for the ākonga. This provides them with ways to communicate with each other while working in small groups and participating in the whole-class discussion.

Kaiako might do this by highlighting particular ideas, adopting ākonga language, asking them to describe key aspects in their own words, and asking them to restate what others have said.

In the haircutting problem, what language might be confusing or unfamiliar to some or all ākonga? How could you support ākonga to develop common language to describe key aspects of this task?

Maintain the cognitive demand

To ensure that the cognitive demand of the task is maintained kaiako should avoid suggesting a particular solution strategy. Kaiako need to help the ākonga to understand the important aspects of the task or problem while leaving solution pathways open. This allows ākonga to reason about significant mathematical ideas both while solving the task and when discussing it at the conclusion.

Anticipating ākonga strategies and planning talk moves to respond to these supports kaiako to avoid the “trap” of leading ākonga to particular solution strategies (and thus shutting down their opportunities for sense-making).

What specifically would you need to avoid during the launch to maintain the cognitive demand of this task?

Conducting high-quality launches requires considerable planning – this cannot be over- emphasised!

Review video #5 Problemand Launch from the series Developing Mathematical Inquiry Communities. Consider how the messages in this video reflect or expand on what you have been reading and thinking about in this topic.

Alison